Permeation modelling

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

Mixed matrix membranes using carbon molecular sieves. II. Modeling permeation behavior./. Membr. Sd., 211 (2), 335-348. 

The starting point for modeling permeation (migration) to the liquid is the second case (ii). This is because it represents the well-studied diffusion of a solute from a polymer of limited volume, VP, into a stirred solution of limited volume, VL. A suitable equation for all of these cases is Eq. (7-51), where cP0 = cPe. 

In this paragraph shortly the permeation measurement method is introduced, followed by various examples of permeation through a silicalite-1 membrane on a sintered stainless steel support. This includes unary and binary mixtures as a function of partial pressure, composition and temperature. Finally the present state of modelling permeation through silicalite-1 membranes is reviewed. 

Geraldes and de Pinho [37] modeled NF permeate using a prepared salt solution to evaluate the feasibility of ED for the removal of chloride ions. The modeled permeates contained chloride ions from 1650 to 2250 mg/L depending on the water recovery. In that study, the quality requirements for the washing waters for pulp bleaching were 350 mg/L of NaCl and 30 PtCo units of color. The NF followed by ED met these requirements and the evaluated costs were 0.65 per m at 80% water recovery. Rapp et al. [131] reported the electrical power consumption of ED to be 0.97 kW h/kg chloride removed. 

Twist JN, Zatz JL. Membrane-solvent-solute interaction in a model permeation system. J Pharm Sci 1988 Jun 77(6) 536-540. 

Solution-diffusion model In the solution-diffusion model, permeates dissolve in the membrane material and then diffuse through the membrane down a concentration gradient. Separation is achieved between different permeates because of differences in the amount of material that dissolves in the membrane and the rate at which the material diffuses through the membrane. The solution-diffusion model is the most widely accepted transport mechanism for many membrane processes [209,210]. Selectivity and permeability of a pervaporation membrane mainly depend on the first two steps, that is, the solubility and diffusivity of the components in the membrane. According to this model, mass transport can be divided into the three steps the mechanism is shown in Fig. 3.11 ... 

Pore-flow model In pore-flow model, permeates are separated by pressure-driven convective flow through tiny pores. Separation is achieved between different permeates because... 

Modeling permeation through porous media Model inputs... 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

In general, the rate of permeation of the permeating species is difficult to calculate. It is a complex matter which intimately involves a knowledge of the structure and dynamics of the membrane and the structure and dynamics of the complex fluid mixture in contact with it on one side and the solvent on the other side. Realistic membranes with realistic fluids are beyond the possibihties of theoretical treatment at this time. The only way of dealing with anything at all reahstic is by computer simulation. Even then one is restricted to rather simplified models for the membrane. 

Most GPC columns are provided with vendor estimates of the plate count of the column and a chromatogram of a series of test peaks. These plate count estimates are usually obtained using small molecule analytes that elute at the total permeation volume (Vp) of the column. The Gaussian peak shape model... 

This review addresses the issues of the chemical and physical processes whereby inorganic anions and cations are selectively retained by or passed through cell membranes. The channel and carrier mechanisms of membranes permeation are treated by means of model systems. The models are the planar lipid bilayer for the cell membrane, Gramicidin for the channel mechanism, and Valinomycin for the carrier mechanism. 

It should be apparent that the principles of selective ion transport are independent of the specific models being treated here and that many of these principles are at variance with what were traditional views on the basis of selective membrane permeation by inorganic ions. Thus, the concept of selectivity among monovalent cations being based on values of hydrated radii is replaced by the... 

It is not possible at present to provide an equation, or set of equations, that allows the prediction from fu st principles of the membrane permeation rate and solute rejection for a given real separation. Research attempting such prediction for model systems is underway, but the physical properties of real systems, both the membrane and the solute, are too complex for such analysis. An analogous situation exists for conventional filtration processes. The general... 

ADMET polymers are easily characterized using common analysis techniques, including nuclear magnetic resonance ( H and 13C NMR), infrared (IR) spectra, elemental analysis, gel permeation chromatography (GPC), vapor pressure osmometry (VPO), membrane osmometry (MO), thermal gravimetric analysis (TGA), and differential scanning calorimetry (DSC). The preparation of poly(l-octenylene) (10) via the metathesis of 1,9-decadiene (9) is an excellent model polymerization to study ADMET, since the monomer is readily available and the polymer is well known.21 The NMR characterization data (Fig. 8.9) for the hydrogenated versions of poly(l-octenylene) illustrate the clean and selective nature of ADMET. 

Attempts were made to determine number average molecular weights (Mn) by osmometry (Mechrolab Model 502, high speed membrane osmometer, 1 to 10 g/1 toluene solution at 37 °C), however, in many instances irreproducible data were obtained, probably due to the diffusion of low molecular weight polymer through the membrane. This technique was abandoned in favor of gel permeation chromatography (GPC). 

Gel Permeation Chromatography. The instrument used for GPC analysis was a Waters Associates Model ALC - 201 gel permeation chromatograph equipped with a R401 differential refractometer. For population density determination, polystyrene powder was dissolved in tetrahydrofuran (THF), 75 mg of polystyrene to SO ml THF. Three y -styragel columns of 10, 10, 10 A were used. Effluent flow rate was set at 2.2 ml/min. Total cumulative molar concentration and population density distribution of polymeric species were obtained from the observed chromatogram using the computer program developed by Timm and Rachow (16). 

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